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Multilinear Harmonic Analysis and Applications

多线性谐波分析及应用

基本信息

批准号：
2154356
负责人：
Polona Durcik
金额：
$ 11.08万
依托单位：
Chapman University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154356&HistoricalAwards=false
关键词：
Multilinear Harmonic Analysis Applications

项目摘要

Harmonic analysis is concerned with decomposing signals or functions into elementary waves, analyzing the pieces individually, and then synthesizing this local information into information about the original object. This approach has wide applications in physics and engineering, but also within mathematics itself. A special focus of harmonic analysis is to obtain good quantitative information on the objects in question, which can often be applied to yield strong quantitative results in the fields of ergodic theory and combinatorics. Ergodic theory investigates behavior of dynamical systems that evolve for a long time, while one focus of combinatorics is to study existence of structures in large but otherwise arbitrary sets. These structures can be geometric patterns such as arithmetic progressions, vertices of a square, their translates, rotates, or dilates. The project will focus on the study of integral transformations that naturally occur within harmonic analysis and build a set of tools that further expands the reach of harmonic analysis into the nearby fields of ergodic theory and combinatorics. As these are very foundational subjects, the results and techniques from these areas are applicable in a wider scientific context. Examples include applications in stochastic analysis, partial differential equations, signal processing, and medical imaging. The project’s fundamental mathematical research will address the community’s currently relevant scientific questions in its continued pursuit in understanding open mathematical questions. The project will further entail working with a postdoc and mentoring undergraduate students, organization of seminars and workshops that promote cross-collaboration and exchange of ideas. The first part of this project will deal with a fundamental family of multilinear singular integral forms with a hypergraph structure. The main goal will be to address boundedness of these forms on Lebesgue spaces. These questions will be approached by extending the techniques previously developed by the PI and collaborators. The work is partially inspired by a major open question in harmonic analysis, boundedness of the simplex Hilbert transforms. Further, this project will address multilinear oscillatory integral forms and their connections with variants of multilinear singular integrals and maximal functions with curvature. Estimates on multilinear singular and oscillatory integrals will then be applied to ergodic theory. In this context, the main goal will be to investigate convergence of various types of ergodic averages along the orbits of commuting measure-preserving transformations by establishing qualitative convergence statements via stronger quantitative variational estimates. The latter will be translated to questions in Euclidean spaces and addressed with harmonic analysis techniques developed throughout this project. These methods will also be applied to questions in Euclidean Ramsey theory, where existence of linear and non-linear point configurations in large subsets of the Euclidean space will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

谐波分析涉及将信号或函数分解为基本波，单独分析这些部分，然后将这些局部信息合成为有关原始对象的信息。这种方法在物理学和工程学中有广泛的应用，但也在数学本身。调和分析的一个特别关注点是获得有关对象的良好定量信息，这通常可以在遍历理论和组合数学领域中产生强有力的定量结果。遍历理论研究的是长期演化的动力系统的行为，而组合学的一个焦点是研究大的但任意的集合中结构的存在性。这些结构可以是几何图案，例如算术级数、正方形的顶点、它们的平移、旋转或膨胀。该项目将重点研究调和分析中自然发生的积分变换，并建立一套工具，进一步将调和分析的范围扩展到遍历理论和组合学的附近领域。由于这些都是非常基础的主题，这些领域的成果和技术适用于更广泛的科学背景。实例包括随机分析、偏微分方程、信号处理和医学成像中的应用。该项目的基础数学研究将解决社区目前相关的科学问题，以继续追求理解开放的数学问题。该项目还将需要与博士后合作，指导本科生，组织研讨会和讲习班，促进交叉合作和思想交流。该项目的第一部分将处理具有超图结构的多线性奇异积分形式的基本族。主要目标是解决这些形式在勒贝格空间上的有界性。这些问题将通过扩展PI和合作者先前开发的技术来解决。这项工作的部分灵感来自调和分析中的一个主要的开放问题，单纯形希尔伯特变换的有界性。此外，这个项目将解决多线性振荡积分形式及其与多线性奇异积分和曲率极大函数的变体的连接。估计多线性奇异和振荡积分，然后将应用到遍历理论。在这种情况下，主要目标将是调查收敛的各种类型的遍历平均沿着轨道的交换措施保持转换通过建立定性收敛声明，通过更强的定量变分估计。后者将被翻译成欧几里德空间中的问题，并在整个项目中开发的谐波分析技术解决。这些方法也将被应用到欧几里德拉姆齐理论的问题，在那里存在的线性和非线性点配置在大子集的欧几里德空间将被调查。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。